For the second part, we have to show that \(R\)  is not a BL-ring. It is shown in [1]} that the divisibility property of BL-rings is equivalent to the one which makes a ring a multiplication ring. Now suppose that \(R\)  is a BL-ring. Then \(R\)  is a multiplication ring. Since \(R\)  is local, then \(R\)  is principal. This contradicts the fact that \(R\)  is a non-noetherian ring.
\(\square \)
